There is a difference between a fishing shelter and an ice-fishing 'house'. The shelter is meant to be moved and is classified as a "portable canopy" or "portable shelter". The key benefits of any ice fishing shelter is simply two things:
Shelter from the wind
Keep heat inside
There are several styles such as:
You can build your own but purchasing one from a manufacturer will make your life a little easier. You can find them from different dealers.
A small portable canopy will work very well for your ice fishing. Most any of the small, pop-up canopies would be perfect as long as you get the walls that can be purchased as an accessory to keep the wind off you.